Lect9 - Chapter 2. Determinants Math1111 Determinants...

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Unformatted text preview: Chapter 2. Determinants Math1111 Determinants Further Properties (Cont’d) Theorem Let E be an elementary matrix. Then det ( EA ) = ( det E )( det A ) . Chapter 2. Determinants Math1111 Determinants Further Properties (Cont’d) Theorem Let E be an elementary matrix. Then det ( EA ) = ( det E )( det A ) . Moreover, det E =     - 1 if E is of type I, α 6 = if E is of type II, 1 if E is of type III. Chapter 2. Determinants Math1111 Determinants Further Properties (Cont’d) Theorem Let E be an elementary matrix. Then det ( EA ) = ( det E )( det A ) . Moreover, det E =     - 1 if E is of type I, α 6 = if E is of type II, 1 if E is of type III. Corollary Let E be an elementary matrix. Then det ( AE ) = ( det E )( det A ) . Chapter 2. Determinants Math1111 Determinants Further Properties (Cont’d) Theorem Let E be an elementary matrix. Then det ( EA ) = ( det E )( det A ) . Moreover, det E =     - 1 if E is of type I, α 6 = if E is of type II, 1 if E is of type III. Corollary Let E be an elementary matrix. Then det ( AE ) = ( det E )( det A ) . Proof. Exercise. Chapter 2. Determinants Math1111 Determinants Further Properties (Cont’d) Proof. Type I case: let E ij be the elementary matrix obtained by interchanging the i th and j th rows of I . Then E ij A is obtained by interchanging the i th & j th rows of A . Chapter 2. Determinants Math1111 Determinants Further Properties (Cont’d) Proof. Type I case: let E ij be the elementary matrix obtained by interchanging the i th and j th rows of I . Then E ij A is obtained by interchanging the i th & j th rows of A . If A =                a 1 1 a 1 2 ··· a 1 n . . . . . . . . . . . . a i 1 a i 2 ··· a i n . . . . . . . . . . . . a j 1 a j 2 ··· a j n . . . . . . . . . . . . a n 1 a n 2 ··· a n n                , then E ij A =                a 1 1 a 1 2 ··· a 1 n . . . . . . . . . . . . a j 1 a j 2 ··· a j n . . . . . . . . . . . . a i 1 a i 2 ··· a i n . . . . . . . . . . . . a n 1 a n 2 ··· a n n                . Chapter 2. Determinants Math1111 Determinants Further Properties (Cont’d) Proof. Type I case: let E ij be the elementary matrix obtained by interchanging the i th and j th rows of I . Then E ij A is obtained by interchanging the i th & j th rows of A . If A =                a 1 1 a 1 2 ··· a 1 n . . . . . . . . . . . . a i 1 a i 2 ··· a i n . . . . . . . . . . . . a j 1 a j 2 ··· a j n . . . . . . ....
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This note was uploaded on 09/06/2010 for the course MATH MATH1111 taught by Professor Forgot during the Fall '08 term at HKU.

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Lect9 - Chapter 2. Determinants Math1111 Determinants...

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